$ B = \left[\begin{array}{rrr}3 & 3 & 0 \\ 1 & 0 & -2\end{array}\right]$ $ C = \left[\begin{array}{rr}1 & 2 \\ 2 & -2 \\ 3 & -2\end{array}\right]$ What is $ B C$ ?
Because $ B$ has dimensions $(2\times3)$ and $ C$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ B C = \left[\begin{array}{rrr}{3} & {3} & {0} \\ {1} & {0} & {-2}\end{array}\right] \left[\begin{array}{rr}{1} & \color{#DF0030}{2} \\ {2} & \color{#DF0030}{-2} \\ {3} & \color{#DF0030}{-2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ B$ , with the corresponding elements in column $j$ of the second matrix, $ C$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ B$ with the first element in ${\text{column }1}$ of $ C$ , then multiply the second element in ${\text{row }1}$ of $ B$ with the second element in ${\text{column }1}$ of $ C$ , and so on. Add the products together. $ \left[\begin{array}{rr}{3}\cdot{1}+{3}\cdot{2}+{0}\cdot{3} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ B$ with the corresponding elements in ${\text{column }1}$ of $ C$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{1}+{3}\cdot{2}+{0}\cdot{3} & ? \\ {1}\cdot{1}+{0}\cdot{2}+{-2}\cdot{3} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ B$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ C$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{1}+{3}\cdot{2}+{0}\cdot{3} & {3}\cdot\color{#DF0030}{2}+{3}\cdot\color{#DF0030}{-2}+{0}\cdot\color{#DF0030}{-2} \\ {1}\cdot{1}+{0}\cdot{2}+{-2}\cdot{3} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{3}\cdot{1}+{3}\cdot{2}+{0}\cdot{3} & {3}\cdot\color{#DF0030}{2}+{3}\cdot\color{#DF0030}{-2}+{0}\cdot\color{#DF0030}{-2} \\ {1}\cdot{1}+{0}\cdot{2}+{-2}\cdot{3} & {1}\cdot\color{#DF0030}{2}+{0}\cdot\color{#DF0030}{-2}+{-2}\cdot\color{#DF0030}{-2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}9 & 0 \\ -5 & 6\end{array}\right] $